Why Adjust Bootstrap Quantiles?
A bootstrap interval is built from the distribution of estimates across resampled datasets. The percentile interval uses quantiles of that distribution directly; the basic interval reflects those quantiles around the original estimate. A bias-corrected and accelerated (BCa) interval instead adjusts the quantile probabilities before taking bootstrap order statistics. The adjustments account for two features: a shift in the bootstrap estimates relative to the original estimate, and asymmetry in how the estimator’s variability changes across the parameter’s values.
In population pharmacokinetics (PK), this can matter when a parameter such as clearance or a random-effects standard deviation has a skewed sampling distribution. The BCa calculation does not repair a poor resampling design, unstable model fitting, or inadequate data. It gives an adjusted interval only when the bootstrap and jackknife calculations are meaningful for the estimator and sampling units.
Bias Correction and Acceleration
The bias-correction value measures the location of \(\hat{\theta}\) within the bootstrap distribution. With no ties, let \(r\) be the number of bootstrap estimates below \(\hat{\theta}\). Then \(z_0=\Phi^{-1}(r/B)\), where \(\Phi\) is the standard normal cumulative distribution function. A positive \(z_0\) means that more than half the bootstrap estimates fall below the observed estimate; a negative value means fewer than half do. If there are ties, a common convention counts each tie as one-half when calculating the proportion. The convention used should be reported.
The acceleration is calculated from delete-one jackknife estimates. Let \(\hat{\theta}_{(i)}\) be the estimate after removing resampling unit \(i\), and let \(\bar{\theta}_{(.)}\) be the mean of the \(n\) delete-one estimates. For independent subject-level data, the subject is the delete-one unit. If the bootstrap resamples clusters, the jackknife should delete clusters, not individual records within them.
For a two-sided interval with nominal coverage \(1-\alpha\), define \(z_{\alpha/2}=\Phi^{-1}(\alpha/2)\) and \(z_{1-\alpha/2}=\Phi^{-1}(1-\alpha/2)\). The adjusted lower and upper probabilities are
The interval endpoints are the \(q_{\mathrm{low}}\) and \(q_{\mathrm{high}}\) quantiles of the bootstrap estimates. For a discrete set of \(B\) estimates, this tutorial uses the nearest-rank rule: sort the estimates and take the order statistic at rank \(\lceil Bq\rceil\). Other software may interpolate between order statistics, so the quantile convention should be stated.
Why the Adjustments Work
The bias correction recenters the normal-score scale according to where the observed estimate lies in the bootstrap distribution. If the bootstrap distribution is centered so that half the estimates lie below \(\hat{\theta}\), then \(r/B\) is about \(0.5\), and \(z_0\) is about zero. If the distribution is shifted, the adjusted probabilities move away from the nominal tails to account for that shift.
The acceleration term reflects skewness in the influence of individual sampling units on the estimator. A short justification comes from the jackknife deviations \(d_i=\bar{\theta}_{(.)}-\hat{\theta}_{(i)}\). Their normalized third moment is \(\sum d_i^3/(\sum d_i^2)^{3/2}\); the factor \(1/6\) is the standard scaling in the BCa adjustment. If the jackknife deviations are symmetric around zero, their cubed deviations cancel and \(a=0\). If influential omissions pull the estimate more strongly in one direction, the cubic sum is nonzero and the two adjusted tails can move differently. Thus \(z_0\) concerns location in the bootstrap distribution, whereas \(a\) captures a local asymmetry related to how sampling variation changes with the parameter.
BCa is typically motivated as a second-order accurate interval under regularity conditions for a smooth estimator and a well-behaved sampling distribution. In practical PK analyses, this is an approximation, not a guarantee of nominal coverage. The resampling units must represent independent sampling units under the design; the fitted model and estimator must be sufficiently stable; and the jackknife estimates must exist and vary smoothly. Boundary constraints, very sparse data, multimodality, or a large fraction of failed fits can undermine these conditions.
Worked Example: A 95% BCa Interval for Clearance
Suppose a hypothetical subject-level bootstrap of a population clearance estimate produces \(B=1000\) converged estimates. Exactly 600 estimates are below the original estimate, with no ties, so \(r/B=0.6000\) and \(z_0=\Phi^{-1}(0.6000)\approx0.2533\). A delete-one-subject jackknife gives deviations from their mean of \(-0.02,\ 0.01,\ 0.01\) L/h.
First calculate the acceleration. The units cancel because the numerator has units \((\mathrm{L/h})^3\) and the denominator also has units \((\mathrm{L/h})^3\):
For a 95% interval, the nominal normal scores are approximately \(-1.960\) and \(1.960\). Substituting \(z_0=0.2533\) and \(a=-0.06803\) gives:
The nearest-rank positions are \(\lceil1000(0.0467)\rceil=47\) and \(\lceil1000(0.9853)\rceil=986\). Suppose the sorted hypothetical bootstrap estimates at those ranks are 1.08 L/h and 2.12 L/h. The BCa interval is therefore \([1.08,\ 2.12]\) L/h. These endpoints are the specified order statistics, not values inferred from the adjusted probabilities alone. The interval is compatible with adjustment in both tails: the lower probability is below 0.025 only if the full adjustment dictates it, and here it is about 0.0467; the upper is about 0.9853.
The interpretation is that this procedure yields a 95% BCa confidence interval under its assumptions and resampling scheme. It does not mean there is a 95% probability that the fixed population clearance lies in this particular computed interval.
Worked Examples: Reading the Adjustments
Worked Example: When Both Adjustments Are Zero
Consider a hypothetical 90% interval for a population volume parameter, using \(B=200\) bootstrap estimates. Suppose 100 estimates are below the original estimate, none are tied, and the delete-one jackknife deviations are \(-0.02,\ 0,\ 0.02\) L. Then \(z_0=\Phi^{-1}(100/200)=0\). The cubed deviations sum to zero, so \(a=0\).
With \(z_{\alpha/2}=-1.6449\) and \(z_{1-\alpha/2}=1.6449\), the adjustment formula gives \(q_{\mathrm{low}}=\Phi(-1.6449)=0.0500\) and \(q_{\mathrm{high}}=\Phi(1.6449)=0.9500\). Under the nearest-rank rule, the ranks are \(\lceil200(0.05)\rceil=10\) and \(\lceil200(0.95)\rceil=190\). If the estimates at those ranks are 4.60 L and 5.40 L, the BCa interval is \([4.60,\ 5.40]\) L. In this special case the BCa quantiles match the nominal percentile quantiles.
Worked Example: Positive Bias Correction and Acceleration
Suppose a hypothetical 95% bootstrap for a population parameter has \(B=2000\) replicates, of which 1260 are below \(\hat{\theta}\). Then \(r/B=0.6300\), giving \(z_0=\Phi^{-1}(0.6300)\approx0.3319\). Assume the delete-one jackknife calculation gives \(a=0.0400\). These values indicate a positive bias-correction score and positive acceleration; they do not by themselves determine the interval endpoints.
For the lower tail, \(z_0+z_{\alpha/2}=0.3319-1.960=-1.6281\). Thus the adjusted normal score is approximately \(0.3319-1.6281/[1-0.0400(-1.6281)]=-1.197\), giving \(q_{\mathrm{low}}\approx0.116\). For the upper tail, \(z_0+z_{1-\alpha/2}=2.2919\), and the adjusted score is approximately \(0.3319+2.2919/[1-0.0400(2.2919)]=2.855\), giving \(q_{\mathrm{high}}\approx0.9978\).
The nearest ranks are \(\lceil2000(0.116)\rceil=232\) and \(\lceil2000(0.9978)\rceil=1996\). If the hypothetical sorted estimates at those ranks are 1.72 and 3.08 in the parameter’s units, the interval is \([1.72,\ 3.08]\) in those units. The strongly separated adjusted probabilities illustrate how the two corrections can change the tails unequally. The upper tail is near the extreme of the finite bootstrap distribution, so its endpoint is especially sensitive to the replicate count and quantile convention.
Assumptions, Diagnostics, and Alternatives
Before reporting BCa limits, confirm that the resampling matches the study design. A subject-level bootstrap is appropriate when subjects are the independent sampling units; repeated records from one subject should not be treated as independent resampling units. Stratified or clustered designs require resampling and jackknife calculations that respect those structures. The estimator must be recalculated consistently in every replicate, including the same parameterization and fitting rules. Whether parameters are refitted or held fixed defines a different uncertainty target; BCa corrections cannot make those targets interchangeable.
Check the following features before interpreting the interval:
- Bootstrap distribution: inspect its shape and the proportion of estimates below \(\hat{\theta}\). A proportion near zero or one makes \(z_0\) extreme or undefined; ties can materially affect the correction.
- Jackknife stability: inspect the delete-one estimates and the sum of squared deviations. If that sum is zero, the acceleration formula is undefined. A few extreme omissions can dominate the cubic sum and produce unstable \(a\).
- Adjusted probabilities: verify that the denominators \(1-a(z_0+z_\alpha)\) are nonzero and that adjusted probabilities lie strictly between zero and one. Very extreme probabilities may request order statistics unsupported by the available \(B\).
- Fit behavior: report convergence failures and investigate whether failures are concentrated in particular resamples. Silently discarding failed replicates can distort both the bootstrap distribution and \(z_0\).
- Parameter scale and constraints: a parameter near a boundary, such as a variance component near zero, may have a nonregular distribution for which standard BCa behavior is unreliable.
Compared with the percentile interval, BCa uses adjusted rather than nominal tail probabilities. Compared with the basic interval, it is not formed by reflecting bootstrap quantiles around \(\hat{\theta}\). These alternatives are useful reference intervals, but their differing formulas do not imply that one must always be wider or more reliable. A parametric bootstrap is another possible resampling approach when the fitted population model is used to generate new datasets; choosing that generator is separate from choosing BCa as the interval construction.
BCa has important limitations. It is a scalar interval construction and does not, by itself, describe dependence among several parameter estimates. With a finite number of replicates, tail order statistics are discrete and may be noisy, especially for extreme adjusted probabilities. If the model is nonidentifiable, estimates are multimodal, or a parameter is commonly on a boundary, the smooth-estimator justification may fail. In such cases, compare interval methods, examine the resampling distribution, and state the limitation rather than treating BCa as an automatic correction.
Check Your Understanding
Answer each question, then compare with the solutions below.
- If 420 of 800 bootstrap estimates fall below \(\hat{\theta}\), with no ties, what is \(r/B\), and is \(z_0\) positive or negative?
- What does \(a=0\) imply about the cubic sum of jackknife deviations? Does it prove the bootstrap distribution is symmetric?
- For \(B=500\) and an adjusted probability of \(0.08\), what order-statistic rank is used under the nearest-rank rule?
- Name two diagnostics that are especially important when a BCa interval is computed for a PK variance parameter near zero.
Solutions. (1) \(r/B=420/800=0.525\), so \(z_0=\Phi^{-1}(0.525)\) is slightly positive. (2) The cubic deviations sum to zero, assuming the denominator is nonzero. This describes the jackknife deviations, not necessarily symmetry of the full bootstrap distribution. (3) The rank is \(\lceil500(0.08)\rceil=\lceil40\rceil=40\). (4) Inspect the bootstrap mass at or near the boundary and the delete-one estimates and acceleration for instability; also check convergence failures and whether the smooth-estimator assumptions are plausible.